February, 1999
Laurent Calvet,and Adlai Fisher
ABSTRACT
This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time
process, the Poisson multifractal. Out model captures the thick tails and volatility persistence exhibited by many
financial time series. We assume that the forecaster knows the true generating process with certainty, but only
observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension
of the state space. We show that a discretized version of the model has a finite state space, which allows an analytical
solution to the conditioning problem. Further, the discrete model converges to the continuous-time model as time
scale goes to zero, so that forecasts are consistent. The methodology is implemented on simulated data calibrated
to the Deutschemark/US Dollar exchange rate. Applying these results to option pricing, we find that the model captures
both volatility smiles and long-memory in the term structure of implied volatilities.
Subject: Investments/Volatility; Investments/Econometrics; Investments/Derivatives
Classification: Empirical/Theory
Calvet: lcalvet@kuznets.harvard.edu
Fisher: (212) 998-0304 afisher@stern.nyu.edu
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